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title: Unit 01 notesAll outcomes are events. An event is usually a partial description. Outcomes are events given with a complete description.
Here ‘complete’ and ‘partial’ are within the context of the probability model.
Notice:
When an event happens, the fact that it has happened constitutes information.
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Using this notation, we can consider an outcome itself as an event by considering the “singleton” subset
Flip a fair coin two times and record both results.
Outcomes: sequences, like
Sample space: all possible sequences, i.e. the set
Events: for example:
With this setup, we may combine events in various ways to generate other events:
Complex events: for example:
Flip a fair coin five times and record the results.
How many elements are in the sample space? (How big is
How many events are there? (How big is
Given two events
They are mutually exclusive when
They are collectively exhaustive
Algebraic rules
Associativity:
Distributivity:
De Morgan’s Laws
In other words: you can distribute “
A probability measure is a function
Kolmogorov Axioms:
Axiom 1:
(probabilities are not negative!)
Axiom 2:
(probability of “anything” happening is 1)
Axiom 3: additivity for any countable collection of mutually exclusive events:
A probability model or probability space consists of a triple
It is a consequence of the Kolmogorov Axioms that additivity also works for finite collections of mutually exclusive events:
A probability measure satisfies these rules.
They can be deduced from the Kolmogorov Axioms.
Negation: Can you find
Monotonicity: Probabilities grow when outcomes are added:
Inclusion-Exclusion: A trick for resolving unions:
The principle of inclusion-exclusion generalizes to three events:
The same pattern works for any number of events!
The pattern goes: “include singles” then “exclude doubles” then “include triples” then ...
Include, exclude, include, exclude, include, ...
The professor chooses three students at random for a game in a class of 40, one to be Host, one to be Player, one to be Judge. What is the probability that Lucia is either Host or Player?
&&& Set up the probability model.
Label the students
Outcomes: assignments such as
These are ordered triples with distinct entries in
Sample space:
Events: any subset of
Probability measure: assume all outcomes are equally likely, so
In total there are
Therefore
Therefore
&& Define the desired event.
Want to find
Define
So we seek
&&& Compute the desired probability.
Importantly,
There are no outcomes in
By additivity, we infer
Now compute
There are
Therefore
Therefore:
Now compute
Finally compute that
At Mr. Jefferson’s University, 25% of students have an iPhone, 30% have an iPad, and 60% have neither.
What is the probability that a randomly chosen student has some iProduct? (Q1)
What about both? (Q2)
&&& Set up the probability model.
A student is chosen at random: an outcome is the chosen student.
Sample space
Write
All students are equally likely to be chosen: therefore
Therefore
Furthermore,
& Define the desired event.
Q1:
Q2:
&&& Compute the probabilities.
We do not believe
Try: apply inclusion-exclusion:
We know
Notice the complements in
Negation:
Try again: Negation:
And De Morgan (or a Venn diagram!):
Therefore:
We have found Q1:
Applying the RELATION from inclusion-exclusion, we get Q2:
The conditional probability of “
This conditional probability
By letting the actuality of event
It is possible to verify each of the Kolmogorov axioms for this function, and therefore
What does it really mean?
Conceptually,
Mathematically,
The definition of conditional probability can also be turned around and reinterpreted:
“The probability of
This principle generalizes to any events in sequence:
The generalized rule can be verified like this. First substitute
Flip a fair coin 4 times and record the outcomes as sequences, like
Let
First let’s calculate
Define
Therefore,
Now suppose we find out that “at least one heads definitely came up”. (Meaning that we know
Now what is our estimate of likelihood of
The formula for conditioning gives:
Therefore:
Flip a coin. If the outcome is heads, roll two dice and add the numbers. If the outcome is tails, roll a single die and take that number. What is the probability of getting a tails AND a number at least 3?
This “two-stage” experiment lends itself to a solution using the multiplication rule for conditional probability.
Two cards are drawn from a standard deck (without replacement).
What is the probability that the first is a 3, and the second is a 4?
This “two-stage” experiment lends itself to a solution using the multiplication rule for conditional probability.
For any events
Interpretation: event
This law can be generalized to any partition of the sample space
For a partition
Division into Cases is just the Law of Total Probability after setting
Setup:
Experiment:
What is the probability that the marble you look at is red?
For any events
Start with the observation that
Apply the multiplication rule to each of order:
Equate them and rearrange:
The main application of Bayes’ Theorem is to calculate
Note: these notes use alphabetical order
Assume that 0.5% of people have COVID. Suppose a COVID test gives a (true) positive on 96% of patients who have COVID, but gives a (false) positive on 2% of patients who do not have COVID. Bob tests positive. What is the probability that Bob has COVID?
Some people find the low number surprising. In order to repair your intuition, think about it like this: roughly 2.5% of tests are positive, with roughly 2% coming from false positives, and roughly 0.5% from true positives. The true ones make up only
(This rough approximation is by assuming
If two tests both come back positive, the odds of COVID are now 98%.
If only people with symptoms are tested, so that, say, 20% of those tested have COVID, that is,
There are marbles in bins in a room:
Your friend goes in the room, shuts the door, and selects a random bin, then draws a random marble. (Equal odds for each bin, then equal odds for each marble in that bin.) He comes out and shows you a red marble.
What is the probability that this red marble was taken from Bin 1?
Two events are independent when information about one of them does not change our probability estimate for the other. Mathematically, there are three ways to express this fact:
Events
A collection of events
A potentially weaker condition for a collection
One could also define
Prove that these are logically equivalent statements:
Make sure you demonstrate both directions of each equivalency.
A bin contains 4 red and 7 green marbles. Two marbles are drawn.
Let
(a) With replacement.
(b) Without replacement.
A tree diagram depicts the components of a multi-stage experiment. Nodes, or branch points, represent sources of randomness.
An outcome of the experiment is represented by a pathway taken from the root (left-most node) to a leaf (right-most node). The branch chosen at a given node junction represents the outcome of the “sub-experiment” constituting that branch point. So a pathway encodes the outcomes of all sub-experiments.
Each branch from a node is labeled with a probability number. This is the probability that the sub-experiment of that node has the outcome of that branch.
One can also use a tree diagram to remember quickly how to calculate certain probabilities.
For example, what is
Answer: add up the pathway probabilities (leaf numbers) terminating in
For example, what is
Answer: divide the leaf probability of
Setup:
Experiment:
Questions:
In many “games of chance”, it is assumed by symmetry principles that all outcomes are equally likely. From this assumption we infer the rule for
When this formula applies, it is important to be able to count total outcomes, as well as outcomes satisfying various conditions.
Permutations count the number of ordered lists one can form from some items. For a list of
To see where this comes from:
There are
Combinations count the number of sets (ignoring order) one can form from some items. We define a notation for it like this:
Another name for combinations is the binomial coefficient.
This formula can be derived from the formula for permutations. The possible permutations can be partitioned into combinations: each combination gives a set, and by specifying an ordering of elements in the set, we get a permutation. For a set of
This notation,
There are also ‘higher’ combinations:
The general multinomial coefficient is defined by the formula:
where
The multinomial coefficient measures the number of ways to partition
Notice that
A team of 3 student volunteers is formed at random from a class of 40. What is the probability that Cooper is on the team?
The class has 40 students. Suppose the professor chooses 3 students Wednesday at random, and again 3 on Friday. What is the probability that Haley is chosen today and Hugo on Friday?
A VA license plate has three letters (with no I, O, or Q) followed by four numerals. A random plate is seen on the road.
(a)
(b)